Blow-up Analysis And Compactness Research Of Exponential Nonlinear Problems

In recent years,more and more attention has been paid to the exponential nonlinear problems from differential geometry,mathematical physics and so on.In this paper,we mainly consider blow-up analysis and compactness analysis of exponential nonlinear problems,together with the sharp geometric inequalities,we deeply study the related problems.Firstly,we use convex rearrangement technique and level set estimate to establish the sharp Trudinger–Moser inequality involving N-Finsler-Laplacian operator and L~pnorm perturbation.Moreover,we also obtain the existence of extremal function by blow-up analysis and capacity technique.Secondly,we consider the prescribed curvature equation on Riemannian surface with boundary.Using blow-up analysis method of Liouville equation,together with Trudinger–Moser inequality,it is proved that the energy functional of the corresponding mean field equation has a clear lower bound,on this basis,we give a sufficient condition for existence of solution to the prescribed curvature equation.Then,we establish Lions type concentration-compactness principle of singular Trudinger–Moser inequalities involving N-Finsler-Laplacian operator in bounded do-main.Furthermore,we also obtain the corresponding concentration-compactness prin-ciple on the entire Euclidean space R~N.Next,we consider the Schr¨odinger equation with nonlinearities including critical exponential growth and singular term.By using the minimax method and the concen-tration analysis,together with some refined estimates,the existence of a ground state solution is proved.For the perturbated problem,two distinct nontrivial weak solutions are obtained.Finally,suppose(M,g)be a complete noncompact N-dimensional Riemannian manifold with negative curvature,N?2,we obtain concentration-compactness prin-ciple of singular Trudinger–Moser inequality on M.As an important application,we prove the existence of a ground state solution for a class of elliptic problem on com-plete noncompact Riemannian manifold,we also get a nontrivial weak solution to the perturbated problem.